add support for affine crystals

[mwhansen]

Implementation of affine crystals from classical crystals:

• input is a classical crystal
• an affine crystal can be constructed by providing the methods e0 and f0

Implementation of affine crystals from classical crystal and promotion:

• input is a classical crystal and a promotion operators which corresponds
  to a Dynkin diagram automorphism
• the methods e0 and f0 are computed using the promotion operator

Implementation of Kirillov Reshetikhin crystals:

• Type A_n^{(1)} KR crystals are implemented.
• Type D_n^{(1)}, B_n^{(1)}, A_{2n-1}^{(2)} KR crystals are implemented using plus-minus diagrams
  to construct the promotion operator which corresponds to interchanging nodes 0 and 1
• Type C_n^{(1)} KR crystals are implemented; the methods e0 and f0 are constructed
  using an embedding into the ambient crystal of type A_{2n+1}^{(2)}
• Type A_{2n}^{(2)}, D_{n+1}^{(2)} KR crystals are implemented; the methods e0 and f0 are
  constructed using an embedding into the ambient crystal of type C_n^{(1)} via a similarity
  of crystals

Some documentation links improvements.

Depends on trac ticket #4326 on root systems.

This patch is authored by Brant Jones and Anne Schilling.

CC: @sagetrac-sage-combinat @dwbump

Component: combinatorics

Keywords: affine crystals

Author: Anne Schilling, Brant Jones

Reviewer: Dan Bump

Merged: sage-4.3.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/3663

[anneschilling]

Changed keywords from none to affine crystals

[anneschilling]

comment:2

Attachment: affine-crystal-3663-as.2.patch.gz

[anneschilling]

Attachment: affine-crystal-3663-as.3.patch.gz

[anneschilling]

comment:6

Attachment: affine-crystal-3663-as.4.patch.gz

[anneschilling]

Reviewer: Dan Bump

[anneschilling]

Author: Anne Schilling, Brant Jones

[anneschilling]

improved documentation links added

[anneschilling]

comment:7

Attachment: affine-crystal-3663-as.5.patch.gz

[anneschilling]

Attachment: affine-crystal-3663-as.6.patch.gz

corrected problems with documentation in crystal_morphism

[dwbump]

comment:8

I am reviewing the version of the patch that is in the combinat queue, running under sage 4.1.1.

I ran sage -testall.
The patch introduces no new failures. (Where it appears in the queue there are some failures, but the same failures still occur if you qpop the patch, rebuild and run testall again, so they are not caused by this patch.)

All new methods have docstrings and tests.

Kirillov-Reshetikhin crystals for are crystal bases on modules of quantized enveloping algebras of affine Kac-Moody Lie algebras. They had their origin in the observation that it was sometimes possible to define crystal bases on the data parametrizing the eigenstates in the Bethe Ansatz. Beyond that, they tend to be perfect crystals, from which all integrable modules of the quantum group can be constructed. There is one Kirillov-Reshetikhin crystal B(r,s) based on tableaux of rectangular shape s^r for every positive integer s and index r of the underlying classical crystal.

Constructions of all for the classical untwisted and untwisted types are summarized in Fourier, Schilling and Okado
http://front.math.ucdavis.edu/0811.1604. Most but all of these are implemented in sage by this patch.

The unimplemented crystals are listed here: http://groups.google.com/group/sage-combinat-devel/msg/9571cf3991bca4db?hl=en

I generated quite a few of these and ran C.check() on them. I looked at a few of them more closely. I am confident that the patch is correct. It is also an important advance to have these affine crystals in sage.

Some useful functionality is also added in crystals.py. Namely, morphisms of crystals and some root string operations.

[mwhansen]

Merged: sage-4.3.alpha0